Serway & Jewett — The Laws of Motion: Problem 54

A mobile is formed by supporting four metal butterflies of equal mass \(m\) from a string of length \(L\). The points of support are evenly spaced a distance \(\ell\) apart as shown in Figure P5.54. The string forms an angle \(\theta_1\) with the ceiling at each endpoint. The center section of string is horizontal. (a) Find the tension in each section of string in terms of \(\theta_1\), \(m\), and \(g\). (b) In terms of \(\theta_1\), find the angle \(\theta_2\) that the sections of string between the outside butterflies and the inside butterflies form with the horizontal. (c) Show that the distance \(D\) between the endpoints of the string is \(D = \frac{L}{5} \{2 \cos \theta_1 + 2 \cos[\tan^{-1}(\frac{1}{2} \tan \theta_1)] + 1\}\).